We use mathematical models to study the mechanisms of oscillatory electrical activity arising from ion channels in cell membranes and modulated by intracellular chemical processes. We are interested in both the behavior of single cells and the ways in which cells communicate and modify each other's behavior. Our main application has been to the biophysical basis of insulin secretion in pancreatic beta-cells. We have examined bursting oscillations in membrane potential and the role of electrical coupling between cells in the islet of Langerhans. Long term goals are to understand how the membrane dynamics interact with intracellular events to regulate secretion. We also compare, contrast, and generalize to other secretory cells and neurons, including GnRH-secreting hypothalamic neurons, pituitary somatotrophs, and fast neurotransmitter secretion at nerve terminals. Our primary tool is the numerical solution of ordinary and partial differential equations. We use analytical, geometrical, graphical, and numerical techniques from the mathematical theory of dynamical systems to help construct and interpret the models. Perturbation techniques are used to get analytical results in special cases. We study both detailed biophysical models and simplified models which are more amenable to analysis. Such an approach aids the isolation of the essential or minimal mechanisms underlying phenomena, the search for general principles, and the application of concepts and analogies from other fields. We see a role for our group as intermediaries between the mathematical and biological disciplines. This includes disseminating the insights of mathematical work to biologists in accessible language and alerting mathematicians and other theoreticians to new and challenging problems arising from biological issues. Recent work on this project includes: 1. (Islet calcium and voltage oscillations) We studied the regulation and role of a putative pacemaker channel for islet calcium and membrane potential oscillations. The channel, discovered by Rorsman and colleagues and named Kslow, is a calcium-activated potassium channel of unknown type. We found that this channel is controlled by release of ER calcium as well as calcium influx into the cytosol. We propose that this channel sits in a subspace between a portion of the ER and the plasma membrane. The model suggests specifically that calcium concentration in this subspace is a weighted average of ER and cytosolic calcium. This model was able to account for the inhibition of Kslow by agents that block store uptake, including thapsigargin and, intriguingly, thapsigargin. The model also confirms that this channel can contribute to islet oscillations precisely because it is influenced by the slow kinetics of the ER but other channels, such as the ATP-dependent potassium channel (KATP) may also contribute. (Satin, Bertram, Sherman) (See Goforth et al, 2002). 2.(Computer modeling of calcium diffusion and buffering) We continued development of the CalC ("Calcium Calculator") software package. CalC is designed to simulate the entry of Ca2+ into a presynaptic terminal, and its diffusion, buffering and binding to putative Ca2+-sensitive transmitter release triggers. Among the improvements are: implementation of separate tortuosity functions for Ca2+ and for each of the buffers, support for 2-dimensional cylindrical geometry, and significant improvements in error handling, which make the software more user-friendly. In addition, we ported CalC to the Windows operating system, so that CalC now runs on the Linux, SGI, and Windows/Intel platforms. (Matveev, Sherman) (Software posted at http://mrb.niddk.nih.gov/matveev) 3. (Biophysical modeling of activity-dependent short-term facilitation - STF - of synaptic response.) In collaboration with Prof. Robert Zucker of Berkeley University, we investigated whether the properties of STF observed at the crayfish NMJ could be explained by the accumulation of free Ca2+ at the presynaptic terminal, using a computer modeling approach. To this end, we analyzed a model of calcium-dependent secretion that includes two independent Ca2+ sensors: a secretion sensor responsible for phasic release, located in the high-calcium microdomain region of an open Ca2+ channel, and a higher-affinity Ca2+ sensor responsible for STF, located further away from the channel, and sensitive to the accumulation of lower residual Ca2+. We found that such a model can indeed account for the observed magnitude of STF and its decay time course, as well as the super-linear growth property of STF, if the distance between the STF site and the Ca2+ channel is about 180 nm or more. We also found that, in order for the model to explain the effects of exogenously applied buffers on STF, one has to assume that the diffusion of the exogenous Ca2+ buffers is significantly reduced in the presynaptic terminal, and, moreover, that the buffers are effectively immobilized in the vicinity of the synaptic active zone, presumably due to their interactions with intracellular proteins. (Matveev, Sherman, Zucker) (See Matveev et al, 2002). 4.(Reconstruction of electrical activity in hypothalamic supraoptic magnocellular neurones.) These neurosecretory cells control the release of the hormones oxytocin (OT) and vasopressin (AV), and these in turn directly control such diverse systems as hydration and osmotic stress (AVP and OT); blood pressure (AVP and OT); parturition (OT); and milk release during suckling (OT). Working in close collaboration with in vitro electrophysiologists (Armstrong), we have built a mathematical model that reproduces the in vitro spiking activity and after-potentials that occur in these neurosecretory cells. We have also fit all parameters so that the model reproduces a wide range of experimental protocols. The model shows how currents interact to control the firing rateand the discharge pattern during physiological stresses such as dehydration. Presented in poster form at the 5th International Congress on Neuroendocrinology (Bristol, UK). We have begun a mathematical analysis showing important similarities to as well as differences from the Plant model for bursting in Aplysia R15 neurons. This will be presented at the Society for Neuroscience meeting, November, 2002. Two papers are in preparation. (Roper, Sherman, Armstrong) 5. (Modeling of metabolic insulin signaling.) We developed a detailed model of metabolic insulin signaling, which combines and integrates previous models for specific subsystems, including insulin receptor binding, receptor recycling, and GLUT4 translocation. Key intermediate steps have been added, including phosphorylation of IRS-1, activation of PI3Kinase, production of PIP3, and activation of PKC-zeta and Akt. We used the model to demonstrate that delayed negative feedback through PKC-zeta onto IRS-1 could account for oscillatory transients observed experimentally by Standaert et al in the insulin response. We also showed how the insulin response could be amplified by reducing endogenous negative feedback from protein tyrosine phosphatases, a therapeutic target of current interest for ameliorating insulin resistance.